[seqfan] Lexicographically first 3-free sequence

[Charles Greathouse]

One of the JMM contest winners was Jack Grahl's A229037: Sequence of
positive integers where each is chosen to be as small as possible subject
to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and
k) form an arithmetic progression.

It's clear that a(n) < 2^n, since it suffices to append a number which is
twice as large as the largest term yet appearing to avoid a 3-AP (then use
induction). Can this trivial bound be improved? While at the JMM I had the
opportunity to ask a Ramsey theorist, but she wasn't willing to hazard a
guess on the spot.

The only lower bound is a(n) >= 1, and I assume this cannot be improved,
that is, 1 appears infinitely often. Can this be proved?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University